Upper Bounds on the Relative Entropy and R\'enyi Divergence as a Function of Total Variation Distance for Finite Alphabets

A new upper bound on the relative entropy is derived as a function of the total variation distance for probability measures defined on a common finite alphabet. The bound improves a previously reported bound by Csisz'ar and Talata. It is further extended to an upper bound on the R'enyi divergence of an arbitrary non-negative order (including $\mathrm{\infty }$) as a function of the total variation distance.